A005045 Number of restricted 3 X 3 matrices with row and column sums n.
0, 0, 1, 3, 6, 10, 17, 25, 37, 51, 70, 92, 121, 153, 194, 240, 296, 358, 433, 515, 612, 718, 841, 975, 1129, 1295, 1484, 1688, 1917, 2163, 2438, 2732, 3058, 3406, 3789, 4197, 4644, 5118, 5635, 6183, 6777, 7405, 8084, 8800, 9571, 10383, 11254
Offset: 0
Examples
a(2) = 1: 110 101 011 a(3) = 3: 111 210 210 111 102 111 111 021 012
References
- E. J. Morgan, On 3 X 3 matrices with constant row and column sum, Abstract 763-05-13, Notices Amer. Math. Soc., 26 (1979), page A-27.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- M. F. Hasler, Table of n, a(n) for n = 0..1000
- E. J. Billington (née Morgan) and N. J. A. Sloane, Correspondence, 1978-1991.
- P. Lisonek, Quasi-polynomials: A case study in experimental combinatorics, RISC-Linz Report Series No. 93-18, 1983. (Annotated scanned copy)
- R. J. Mathar, OEIS A005045 [Proof of g.f. for 3 of the 12 cases]
- E. J. Morgan, Construction of Block Designs and Related Results, Ph.D. Dissertation, Univ. Queensland, 1978; Bull. Austral. Math. Soc., Volume 19, Issue 1 August 1978, pp. 139-140.
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
- Index entries for linear recurrences with constant coefficients, signature (2,0,-1,0,-2,2,0,1,0,-2,1).
Crossrefs
Cf. A002817 for another version.
Programs
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Maple
A005045:=-z**2*(-z**5+z**6-z**3+z+1)/((z**2+1)*(z**2+z+1)*(z+1)**2*(z-1)**5); # conjectured by Simon Plouffe in his 1992 dissertation; see formula lines here for the proof of correctness
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Mathematica
a[n_] := Block[{k = Floor[(n + 2)/3]}, Sum[Sum[Sum[If[m + r == i/2, Floor[(n - 2*i + m + 2)/2], n - 2*i + m + 1], {r, 0, Floor[i/2 - m]}], {m, Max[2*i - n, 0], Floor[i/2]}], {i, 1, n - k}]]; Table[a[n], {n, 0, 46}] (* Peter Pein, May 13 2008 *) LinearRecurrence[{2,0,-1,0,-2,2,0,1,0,-2,1},{0,0,1,3,6,10,17,25,37,51,70},50] (* Harvey P. Dale, Nov 15 2018 *) -
PARI
A005045(n)={sum( i=1,n-(n+2)\3, sum( m=max(0,2*i-n),i\2, sum( r=0,i\2-m, if( m+r!=i/2, n-2*i+m+1, (n-2*i+m+2)\2))))} \\ M. F. Hasler, Version 1, May 13 2008
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PARI
A005045(n)={sum( i=1,(2*n)\3, sum( m=max(0,2*i-n),i\2, (n-2*i+m+1)*((i+1)\2-m)+(i%2==0)*(n-2*i+m+2)\2))} \\ M. F. Hasler, Version 2, much faster, May 13 2008
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PARI
concat(vector(2), Vec(x^2*(1 + x - x^3 - x^5 + x^6) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)) + O(x^60))) \\ Colin Barker, Apr 22 2017
Formula
Let n = 3k, 3k-1 or 3k-2 according as n == 0, 2 or 1 mod 3, for n >= 3. Then a(n) = Sum_{i=1..n-k} Sum_{m=max(0,2i-n)..floor(i/2)} Sum_{r=0..floor(i/2)-m} c(i,m,r), where c(i,m,r) = n-2i+m+1 when m+r != i/2, or = floor((n-2i+m+2)/2) when m+r = i/2. [Typos corrected by Peter Pein, May 13 2008]
G.f.: -x^2*(-x^5+x^6-x^3+x+1)/((x^2+1)*(x^2+x+1)*(x+1)^2*(x-1)^5). This was conjectured by Simon Plouffe in his 1992 dissertation and is now known to be correct, although it may be that all the details of the proof have not been written down. See the Mathar link for details.
Extensions
Edited by N. J. A. Sloane, May 12 2008, May 13 2008
More terms from Peter Pein, May 13 2008
Comments